Estimation theory is an influential mathematical concept applied in various communication and signal processing applications. This theory is useful for estimating the desired information in received data and is therefore used in a variety of applications ranging from radar to speech processing. In this section, we will list the basic statistical concepts as well as the estimation process. As we have mentioned, the main mathematical tools used by metrology belong to statistics. In addition, we are particularly interested in estimation theory, which shows how to correctly estimate a quantity from a data sample [1]}. The data can be of any type. For example, the data sample can be a set of the results of random draws, or even the wavelengths of photons coming out of a radioactive sample. Assume that a non-deterministic process produces a set of data \(x=\left\lbrace x_{1},x_{2},...,x_{n}\right\rbrace \) , which is a particular occurrence (or realization) of a set of independent and identically distributed random variables denoted by \(X=\left\lbrace X_{1},X_{2},...,X_{n}\right\rbrace \) . The probability density function corresponding to any random variable in this set is \(p\left(x;\theta \right)\) , where \(x\in R\)  is a possible realization of a random variable and \(\theta \)  is a real number (e.g., it could be a physical parameter, such as the phase in an optical interferometer) that is parametrized by \(p\left(x;\theta \right)\) . Hence, the probability that a measurement result in our data set lies in the interval \(x_{0}\le x\le x_{f}\)  is determined by the integral of \(p\left(x;\theta \right)\)  as : \(\int _{x_{0}}^{x_{f}}p\left(x;\theta \right)dx\) .
